An Effective and Automated Processing of Resonances in Vibrational Perturbation Theory Applied to Spectroscopy

The broader availability of cost-effective methodologies like second-order vibrational perturbational theory (VPT2), also in general-purpose quantum chemical programs, has made the inclusion of anharmonic effects in vibrational calculations easier, paving the way to more accurate simulations. Combined with modern computing hardware, VPT2 can be used on relatively complex molecular systems with dozen of atoms. However, the problem of resonances and their corrections remains a critical pitfall of perturbative methods. Recent works have highlighted the sensitivity of band intensities to even subtle resonance effects, underlying the importance of a correct treatment to predict accurate spectral bandshapes. This aspect is even more critical with chiroptical spectroscopies whose signal is weak. This has motivated the present work in exploring robust methods and criteria to identify resonances not only in energy calculations but also on the transition moments. To study their performance, three molecules of representative sizes ranging from ten to several dozens of atoms were chosen. The impact of resonances, as well as the accuracy achievable once they are properly treated, is illustrated by the changes in spectral bandshapes, including chiroptical spectroscopies.


Transition moments of properties
By expressing the perturbed wave functions into the RSPT development of the transition moment of a property P, the following form is reached, Each line corresponds to a different term in Eq. 8 in the main article. S-4

Resonances in Darling-Dennison terms
As mentioned in the main article, a particularly challenging aspect of the calculation of Darling-Dennison (DD) couplings is that they can be themselves affected by Fermi resonances. If such singularities occur, the corresponding terms would be incorrectly evaluated, with the risk of excessively large values predicted. While this problem is known and has been documented, few strategies have been proposed to identify these resonances, especially in ways compatible with black-box implementations. Indeed, with most studies focused on small molecules, a commonly chosen method is to flag manually resonances, based on the harmonic frequency differences at the denominator, or the overall magnitude of κ A−B hrs . S1-S4 Conversely, ecause of the different nature with respect to the energy, tests like the one proposed by Martin and coworkers cannot be directly applied. While a test like R12CVPT could be used, it cannot be applied on the analytic formulas for the DD terms, but instead must be considered in the development of the equations, so that the resonances are correctly removed. This either requires the development and implementation of different formulas based on the possible resonance cases, or an explicit expansion of the perturbative terms at runtime. S5 Both solutions can represent a significant increase in complexity for the implementation, with the latter having an important computational cost over the use of analytic formulas.
In this work, we have adapted a technique initially designed for the calculation of so-called resonance-free energies, based on the on the degeneracy-corrected PT2 (DCPT2) treatment proposed by Kuhler and coworkers S6 and called the hybrid DCPT2-VPT2 scheme. S7 The principle is similar to Martin's test and designed for the calculation of vibrational energies.
Considering the first-order correction to the energy of two harmonic states, potentially involved in a Fermi resonance and assumed uncoupled from the rest, the variational energy can be written S-5 where k is proportional to the cubic force constant related to the modes involved in states r and s and ∆ = (ε r (0) − ε s (0) )/2. For simplicity, ∆ and k are assumed positive. This can be easily generalized by factorizing the whole term by +1 or -1, using absolute values of these quantities. The square root can then be expanded with respect to k up to the second order, so that the potentially resonant term can be recast as, By choosing appropriately k, the previous equality can be generalized to other quantities, not specific to energy, such as those appearing in the analytic formulas of Darling-Dennison terms. S8 The main limitation of this transformation is that it is valid for negligible couplings (k ∼ 0), and would introduce a significant error compared to the correct term for large values of k far from resonances. To compensate this, an hybridization is operated, by combining the original term (f orig ) with the alternative form obtained through the DCPT2 scheme (f corr ) with a transition function such as an hyperbolic tangent, S7,S8 b controls the transition threshold from the corrected to the original term, and can be set at relatively high value (e.g., b ∼ 10 5 ), while a controls the degree of mixing permitted during the transition.
Step-like functions can be adopted with a = 1. This form can be easily implemented with analytic formulas of Darling-Dennison resonances and have been used here.

Analytic transition moments at the VPT2 level
The transition moments and coefficients associated to 3-quanta states (second overtones, "2+1" binary combinations and "1+1+1" ternary combinations), can be generalized in few equations, by introducing the constant factor, The transitions have then the form, S9-S11 S-8

Coefficients for the second-order peturbed wave function
The coefficient of the second-order wave function involved in the 1-3 Darling-Dennison resonance has the form, S-9 Non-resonant form of the VPT2 transition moments

Fundamental bands
In the case where |1 i ⟩ and |1 j ⟩ are involved in a 1-1 DDR, an alternative formula must be adopted for the transition moment given in Eq. 11.

3-quanta transitions
In the case where |1 i 1 j 1 k ⟩ and |1 l ⟩ are involved in a 1-3 DDR, an alternative formula must be adopted for the transition moment given in Eq. S3, 3.8 Figure S1: Mean absolute error (MAE, upper panel), maximum unsigned error (|MAX|, center panel), and standard deviation (lower panel) for the fundamental energies below 2800 cm −1 for methyloxirane in gas phase (left panels) and neat liquid (right panels), excluding mode 18 (17 modes out of 24). Experimental data were taken from Refs. S12,S13. To facilitate comparisons, the GVPT2 energies were used, selecting the variational overlap with the highest overlap over the fundamental DVPT2 states. R12COEF and R12WFRQ were applied as unique tests, without any energy-relative test.
Resonances were identified with a combination of R12MART/R12COEF (K 1−2 = 1.0) for Fermi resonances and R11HRS/R11COEF (K 1−1 = 10) for 1-1 Darling-Dennison resonances. The variable criteria are specified in the left column.  / cm mol sr Wavenumbers / cm Figure S11: Comparison of theoretical GVPT2 ROA spectra of (S )-2-methyloxirane in neat liquid within the CH-stretching region using different schemes and thresholds (K=K 1−1 I ) for the identification of 1-1 Darling-Dennison resonances. A combination of R12MART (K 1−2 = 1.0) and R12COEF was used for the Fermi resonances, the second test with the thresholds K 1−2 I =0.3 (upper panels), 0.4 (middle panels) and 0.5 (lower panels). Experiment (black dashed lines) was taken from Ref. S13. Gaussian broadening functions with half-width at half-maximum of 10 cm −1 were used to match experiment. The computed spectra were shifted by -18 cm −1 to match experiment.  / cm mol sr Wavenumbers / cm Figure S17: Comparison of theoretical GVPT2 ROA spectra of (S )-2-methyloxirane in neat liquid within the CH-stretching region using different schemes and thresholds (K=K 1−1 I ) for the identification of 1-1 Darling-Dennison resonances. The R12CVPT scheme was used for the Fermi resonances, with the thresholds K 1−2 I =0.1 (upper panels), 0.2 (middle panels) and 0.3 (lower panels). Experiment (black dashed lines) was taken from Ref. S13. Gaussian broadening functions with half-width at half-maximum of 10 cm −1 were used to match experiment. The computed spectra were shifted by -18 cm −1 to match experiment. / dm mol cm Wavenumbers / cm Figure S18: Comparison of theoretical GVPT2 IR spectra of (S )-2-methyloxirane in liquid xenon within the fingerprint region using different schemes and thresholds (K=K 1−1 I ) for the identification of 1-1 Darling-Dennison resonances. A combination of R12MART (K 1−2 = 1.0) and R12COEF was used for the Fermi resonances, the second test with the thresholds K 1−2 I =0.1 (upper panels), 0.2 (middle panels) and 0.3 (lower panels). Experiment (black dashed lines) was taken from Ref. S14. Lorentzian broadening functions with half-width at half-maximum of 2 cm −1 were used to match experiment. The inset shows a zoom of the 1400-1600 cm −1 region.   : Experimental (black dashed lines), S13 harmonic (blue), VPT2 (green) and GVPT2 (red) Raman and ROA spectra of (S )-2-methyloxirane in neat liquid. The automatic procedure used a combination of R12MART/R12COEF (∆ 1−2 = 200 / K 1−2 = 1.0 / K 1−2 I = 0.1) for FRs, R11HRS/R11COEF (∆ 1−1 = 100 / K 1−1 = 10 / K 1−1 I = 0.3) for 1-1 DDRs, R22HRS (∆ 2−2 = 100 / K 2−2 = 20) for 2-2 DDRs. Gaussian broadening functions with half-width at half-maximum of 5 cm −1 were used to match experiment, except in the CH-stretching region, where 10 cm −1 was used. Figure S24: Comparison of theoretical GVPT2 IR spectra of (1R,5R)-α-pinene in CCl 4 within the fingerprint region using different schemes and thresholds (K=K 1−1 I ) for the identification of 1-1 Darling-Dennison resonances. A combination of R12MART (K 1−2 = 1.0) and R12COEF was used for the Fermi resonances, the second test with the thresholds K 1−2 I =0.1 (upper panels), 0.3 (middle panels) and 0.5 (lower panels). Experiment (black dashed lines) was taken from Ref. S16. Lorentzian broadening functions with half-width at half-maximum of 4 cm −1 were used to match experiment. / dm mol cm Wavenumbers / cm Figure S25: Comparison of theoretical GVPT2 IR spectra of (1R,5R)-α-pinene in CCl 4 within the CH-stretching region using different schemes and thresholds (K=K 1−1 I ) for the identification of 1-1 Darling-Dennison resonances. A combination of R12MART (K 1−2 = 1.0) and R12COEF was used for the Fermi resonances, the second test with the thresholds K 1−2 I =0.1 (upper panels), 0.3 (middle panels) and 0.5 (lower panels). Experiment (black dashed lines) was taken from Ref. S16. Lorentzian broadening functions with half-width at half-maximum of 8 cm −1 were used to match experiment. The computed spectra were shifted by -20 cm −1 to match experiment.  Figure S29: Comparison of theoretical GVPT2 IR spectra of (1R,5R)-α-pinene in CCl 4 within the CH-stretching region considering the whole system ("FULL") or excluding the torsional modes related to the methyl groups ("NO2,4,5") with different thresholds (K=K 1−1 I ) for the identification of 1-1 Darling-Dennison resonances. A combination of R12MART (K 1−2 = 1.0) and R12COEF was used for the Fermi resonances, the second test with the thresholds K 1−2 I =0.1 (upper panels), 0.3 (middle panels) and 0.5 (lower panels). Experiment (black dashed lines) was taken from Ref. S16. Lorentzian broadening functions with halfwidth at half-maximum of 8 cm −1 were used to match experiment. The computed spectra were shifted by -20 cm −1 to match experiment. Wavenumbers / cm Figure S30: Comparison of theoretical GVPT2 VCD spectra of (1R,5R)-α-pinene in CCl 4 within the fingerprint region considering thee all system ("FULL") or excluding the torsional modes related to the methyl groups ("NO2,4,5") with different thresholds (K=K 1−1 I ) for the identification of 1-1 Darling-Dennison resonances. A combination of R12MART (K 1−2 = 1.0) and R12COEF was used for the Fermi resonances, the second test with the thresholds K 1−2 I =0.1 (upper panels), 0.3 (middle panels) and 0.5 (lower panels). Experiment (black dashed lines) was taken from Ref. S16. Lorentzian broadening functions with half-width at half-maximum of 4 cm −1 were used to match experiment.  Figure S31: Comparison of theoretical GVPT2 VCD spectra of (1R,5R)-α-pinene in CCl 4 within the CH-stretching region considering thee all system ("FULL") or excluding the torsional modes related to the methyl groups ("NO2,4,5") with different thresholds (K=K 1−1 I ) for the identification of 1-1 Darling-Dennison resonances. A combination of R12MART (K 1−2 = 1.0) and R12COEF was used for the Fermi resonances, the second test with the thresholds K 1−2 I =0.1 (upper panels), 0.3 (middle panels) and 0.5 (lower panels). Experiment (black dashed lines) was taken from Ref. S16. Lorentzian broadening functions with halfwidth at half-maximum of 8 cm −1 were used to match experiment. The computed spectra were shifted by -20 cm −1 to match experiment.  Figure S32: Experimental (black dashed lines) S17 and GVPT2 IR (upper panels) and VCD (lower panels) spectra of artemisinin in chloroform. The models are described in the main text. Lorentzian broadening functions with half-width at half-maximum of 4 cm −1 were used to match experiment.